Uniform Convergence of Regularized Determinants for Zeta Spectral Triples: Toward Global Weil Positivity and the Riemann Hypothesis

This paper reorganizes and polishes a conditional analytic framework for the uniform convergence of regularized determinants associated with zeta spectral triples. The guiding question is precise: under what verifiable analytic hypotheses would the finite self-adjoint spectral triples of Connes, Consani and Moscovici yield a compact-uniform determinant limit equal to Riemann’s completed function? The paper does not claim an unconditional proof of the Riemann Hypothesis. Its main contribution is a carefully normalized theorem: if the recentered minimal eigenvectors of the truncated Weil quadratic forms converge in an exponential weighted topology, if the resulting Mellin transforms are identified with the completed Riemann function after a non-vanishing normalization, and if the determinant normalizations are locally uniform, then the normalized regularized determinants converge uniformly on compact subsets of the complex plane to . Since the finite determinant approximants have only real zeros in the spectral variable, Hurwitz’s theorem then implies the Riemann Hypothesis. The paper also records a rigorous local trace theorem showing that the half-density damped prime-clock trace produces the exact prime-power weights in the Weil explicit formula. The final sections explain how quasi-inner local factor theory, semi-local Sonin spaces, prolate concentration estimates, trace-norm determinant continuity, and Quillen-curvature positivity fit into the remaining convergence problem. The result is a publishable conditional theorem with an explicit analytic bottleneck, rather than an unconditional proof.